Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T05:01:09.719Z Has data issue: false hasContentIssue false
  • English
  • Français

On the nonlinear development of the most unstable Görtler vortex mode

Published online by Cambridge University Press:  26 April 2006

James P. Denier  and
Philip Hall
James P. Denier
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL. UK
Philip Hall
Affiliation:
Department of Mathematics, University of Manchester, Manchester M13 9PL. UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The nonlinear development of the most unstable Görtler vortex mode in boundary-layer flows over curved walls is investigated. The most unstable Görtler mode is confined to a viscous wall layer of thickness O(G−1/5) and has spanwise wavelength O(G−1/5); it is, of course, most relevant to flow situations where the Görtler number G [Gt ] 1. The nonlinear equations governing the evolution of this mode over an O(G−3/5) streamwise lengthscale are derived and are found to be of a fully non-parallel nature. The solution of these equations is achieved by making use of the numerical scheme used by Hall (1988) for the numerical solution of the nonlinear Görtler equations valid for O(1) Görtler numbers. Thus, the spanwise dependence of the flow is described by a Fourier expansion whereas the streamwise and normal variations of the flow are dealt with by employing a suitable finite-difference discretization of the governing equations. Our calculations demonstrate that, given a suitable initial disturbance, after a brief interval of decay, the energy in all the higher harmonics grows until a singularity is encountered at some downstream position. The structure of the flow field as this singularity is approached suggests that the singularity is responsible for the vortices, which are initially confined to the thin viscous wall layer, moving away from the wall and into the core of the boundary layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 247 , February 1993 , pp. 1 - 16
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aihara, Y. 1976 Nonlinear analysis of Görtler vortices. Phys. Fluids 19, 1655.Google Scholar
Benney, D. J. & Chow, K. 1989 Instabilities of waves on a free surface, Stud. in Appl. Maths 74, 227– 243.Google Scholar
Cebeci, T., Khattab, A. A. & Stewartson, K. S. 1981 Three-dimensional boundary layer separation and the ok of accessibility. J. Fluid Mech. 107, 57– 82.Google Scholar
Choudhary, M. & Hall, P. 1992 Small wavenumber Görtler vortices. GJMAM (submitted.)Google Scholar
Denier, J. P., Hall, P. & Seddoughui, S.O. 1990 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. ICASE Rep. 90–31 (referred to herein as DHS).Google Scholar
Denier, J. P., Hall, P., & Seddougui, S.O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness, Phil. Trans. R. Soc Lond. A 335, 51– 85 (referred to herein as DHS.)Google Scholar
Hall, P. 1982a Taylor–Görtler vortices in fully developed or boundary layer flows: linear theory. J. Fluid Mech. 124, 475– 494.Google Scholar
Hall, P. 1982b On the nonlinear evolution of Görtler vortices in growing boundary layers. J.Inst. Maths Applics. 29, 173.Google Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41– 58.Google Scholar
Hall, P. 1984 On the instability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347– 368.Google Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 247– 266.Google Scholar
Hall, P. & Horseman, N. 1991 The inviscid secondary instability of fully nonlinear longitudinal vortex structures in growing boundary layers. J. Fluid Mech. 232, 357– 375.Google Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Proc. R. Soc. Lond. A 415, 421– 444.Google Scholar
Hall, P. & Seddougui, S. O. 1989 On the onset of three-dimensionality and time-dependence in Görtler vortices. J. Fluid Mech. 204, 405– 420.Google Scholar
Prandtl, L. 1935 Aerodynamic Theory. Springer.
Ruban, A. I. 1990 Propagation of wave packets in the boundary layer on a curved surface. Izv. Akad. Nauk, SSSR, Mekh., Zhudk. Gaza 2, 59– 68.Google Scholar
Sabry, A. S. & Liu, J. T. C. 1991 Longitudinal vorticity elements in boundary layers. J. Fluid Mech. 231, 615– 664.Google Scholar
Timoshin, S. N. 1991 Asymptotic analysis of the Görtler vortex spectrum. Fluid Dyn., January 1991, 25.Google Scholar